Koopman embedding and super-linearization counterexamples with isolated equilibria

A linearizing embedding of a nonlinear smooth dynamical system is a global identification of the nonlinear system with an invariant submanifold of a linear dynamical system. Linearizing embeddings have been studied by various communities and are of central importance in the rapidly developing “applied Koopman operator theory” literature [BBKK22]. An oft-repeated claim in that literature is that any dynamical system with multiple isolated equilibria cannot be linearized by a smooth embedding, or at least not by an embedding that “contains the state”. We call the latter type of linearizing embeddings super-linearizations.

The first claim was shown to be false by the authors in [KA23, Ex. 4] if non-Euclidean state spaces are allowed. In the present paper we show that both claims are also false for Euclidean state spaces by constructing for each $k>1$ (i) linearizable dynamical systems on $\mathbb{R}^{k}$ having any countable number of isolated equilibria and (ii) super-linearizable dynamical systems on $\mathbb{R}^{k}$ having any finite number of equilibria. Thus, there are more (super-)linearizable dynamical systems than previously believed.

Super-linearizations are of practical importance for engineering applications since they are invertible in closed form. Our notion of super-linearization is slightly different from that of Belabbas and Chen [BC23] since we consider embeddings into linear rather than affine dynamical systems.

We now proceed more formally. In this paper all manifolds and maps between them are smooth ($C^{\infty}$), and embeddings are smooth embeddings [Lee13]. Let $M$ be a manifold and $\Phi\colon~{}\mathbb{R}\times M\rightarrow M$ be the flow of a dynamical system. A map $f\colon M_{1}\rightarrow M_{2}$ between two such dynamical systems $(M_{1},\Phi_{1})$ and $(M_{2},\Phi_{2})$ is called equivariant (also called a semi-conjugacy) if

for all $t\in\mathbb{R}$. If $f$ is a diffeomorphism then we say that $(M_{1},\Phi_{1})$ and $(M_{2},\Phi_{2})$ are smoothly conjugate dynamical systems.

A dynamical system $(M,\Phi)$ admits a linearizing embedding if there exists an equivariant embedding $f\colon M\to\mathbb{R}^{n}$ of $(M,\Phi)$ into $(\mathbb{R}^{n},\Psi)$ where $\Psi^{t}=\text{exp}(At)$ is the flow of a linear system of ordinary differential equations on $\mathbb{R}^{n}$ generated by some matrix $A$. Recall that $f$ is an embedding if it is a homeomorphism of $M$ onto its image and if the derivative $df$ is injective [Lee13, p. 85].

From now on we shall assume that $M$ is an open subset of $\mathbb{R}^{k}$. For such systems there is a stronger notion of linearizability: a linearizing embedding $f\colon M\rightarrow\mathbb{R}^{k+m}$ is a super-linearizing embedding if it is of the form $f(x)=(x,p(x))$ for some map $p\colon M\rightarrow\mathbb{R}^{m}$. Observe that the image of $f$ is the graph of $p$,

For this reason it will be helpful for us to refer to embeddings $f\colon M\rightarrow\mathbb{R}^{n}$ as graphlike if the image of $f$ can be written in the form

where $N$ is an open subset of some $k$-dimensional subspace of $\mathbb{R}^{n}$ and $\varphi\colon N\rightarrow U$ is a smooth map into a complementary subspace $U$. Equivalently, $f$ is graphlike if and only if there exists a linear subspace $U\subset\mathbb{R}^{n}$ of codimension $k$ whose affine translates transversely intersect the image of $f$ in at most one point. Every super-linearizing embedding is graphlike, and the image of any graphlike linearizing embedding of a dynamical system is the image of a super-linearizing embedding of some smoothly conjugate dynamical system.

The surface $\Sigma$ is not given by a graphlike embedding of $\mathbb{R}^{2}$ into $\mathbb{R}^{3}$. However, it might be possible to find an equivariant embedding of $\mathbb{R}^{3}$ into some higher dimensional vector space whose restriction to $\Sigma$ is graphlike and hence a super-linearizing embedding of some dynamical system. This is the main idea that we use in our proof of Theorem 1. Before presenting the proof we must first discuss equivariant polynomial maps between vector spaces.

Consider a finite-dimensional vector space $V$. The symmetric product $\textnormal{Sym}^{m}V$ can be interpreted as homogeneous degree-$m$ polynomials on the dual $V^{*}$ by taking the evaluation of $\eta\in V^{*}$ on $v_{1}\odot\dots\odot v_{m}$ to be the product

Here $\langle~{},~{}\rangle$ denotes the pairing between $V$ with its dual and $\odot$ is the symmetrized tensor product [FH91, pp. 473–474]. We can then introduce the direct sum

interpreted as the vector space of all degree-$m$ polynomials on $V^{*}$. The diagonal inclusion $V\hookrightarrow\textnormal{Sym}^{m}V$ given by $v\mapsto v\odot\cdots\odot v$ is a $GL(V)$-equivariant degree-$m$ polynomial map, and by extension we may consider the natural map

More explicitly, $\Delta^{m}$ sends $v\in V$ to the polynomial which when evaluated on $\eta\in V^{*}$ yields

Note that $\Delta^{m}$ is a smooth embedding of $V$ into $P^{m}(V^{*})$ and is $GL(V)$-equivariant with respect to the action on polynomials $p$ given by

for $g\in GL(V)$ and where $g^{*}$ is the adjoint. Thus, if $f$ is a linearizing embedding, so is $\Delta^{m}\circ f$.

We will say that the submanifold $\text{Im}(f)$ is tamed by the polynomials $p_{1},\dots,p_{k}$.

Incidentally, the example of $\gamma\subset\mathbb{R}^{2}$ in the previous remark shows that not every embedded submanifold can be tamed by polynomials. If one is interested in applying a polynomial embedding $\Delta^{m}$ to super-linearize a dynamical system, it would be of interest to obtain necessary and sufficient conditions for an embedded submanifold to be tamed by polynomials, and to perhaps bound their degree.

To conclude, we would like to make some comments regarding the relevance of these counterexamples to the study of Koopman eigenmappings. For a dynamical system $(M,\Phi)$ a Koopman eigenfunction is a function $\psi$ on $M$ for which

for all $t$ and for some $\lambda\in\mathbb{R}$. If one has multiple Koopman eigenfunctions $\psi_{1},\dots,\psi_{n}$ then $x\mapsto(\psi_{1}(x),\dots,\psi_{n}(x))$ defines an equivariant map from $M$ into a linear dynamical system on $\mathbb{R}^{n}$. If one has enough functionally independent Koopman eigenfunctions, then this map will be a smooth immersion, and hence, is very closely related to our notion of a linearizing embedding. Indeed, in Example 1 the three functions $x+y$, $x-y$, and $z$ each pull back through the embedding to give Koopman eigenfunctions on a planar dynamical system with multiple isolated equilibria.

As far as we know, these are the first examples of globally defined smooth Koopman eigenfunctions for a dynamical system with multiple isolated equilibria. Furthermore, the linear span of these eigenfunctions seems to be the first known example of a non-trivial finite-dimensional subspace of smooth functions for a dynamical system with multiple isolated equilibria which is invariant under the action of the Koopman operator. Moreover, by using a super-linearizing embedding (as in Example 2) such an invariant subspace additionally “contains the state”.

Finally, we should caution that despite the counterexamples we have presented, there exist many obstructions which prevent the existence of linearizing embeddings for general dynamical systems. The existence of hetero- and homoclinic orbits between equilibria is one such obstruction (since these orbit types are not possible for linear systems), and so too is the presence of multiple asymptotically stable equilibria (see the first footnote in [KA23]). In addition, the main theorem of [LOS23] forbids the existence of linearizing embeddings for systems whose collection of $\omega$-limit set is countable and contains more than one element, and whose forward-orbits are all precompact. We note that this is consistent with our example since it contains unbounded forward-orbits which are not precompact.